How to customize a WordPress Multisite (WPMU) registration form

If you google how to customize a WordPress registration form you’ll only find codex pages instructing how to do that for a single WordPress installation. I find that odd considering how long MU has been out.

Of course I’m not the first to need a custom MU registration form so I eventually found the answer across several forum and stackoverflow posts. I thought I’d collect the complete answer in a single place. Hope it helps.